In this paper we provide a characterization of the nonnegativity of a discrete quadratic functional $$ I with fixed right endpoint in the optimal control setting. This characterization is closely related to the kernel condition earlier introduced by M.Bohner as a part of a focal points definition for conjoined bases of the associated linear Hamiltonian difference system. When this kernel condition is satisfied only up to a certain critical index $$ m , the traditional conditions, which are the focal points, conjugate intervals, implicit Riccati equation, and partial quadratic functionals, must be replaced by a new condition. This new condition is determined to be the nonnegativity of a block tridiagonal matrix, representing the remainder of $$ I after the index $$ m , on a suitable subspace. Applications of our result include the discrete Jacobi condition, a unification of the nonnegativity and positivity of $$ I into one statement, and an improved result for the special case of the discrete calculus of variations. Even when both endpoints of $$ I are fixed, this paper provides a new result.

Last change: October 16, 2002. (c) Roman Hilscher